Question

Consider the general motion of a wheel of radius 2 m which can be viewed as pure translation with the velocity →v and pure rotation about O with angular velocity 5 rad/s. Which of the following is incorrect?

- →VAO=−10^i
- →VBO=−10^j
- →VCO=10^i
- →VDO=−10^j

Solution

The correct option is **D** →VDO=−10^j

As we know,

→VPQ=→ω×→rPQ

So here, →VAO=ω×→rAO

=5(−^k)×2(−^j)=−10^i

As we can see, the direction of velocity w.r.t O will be in negative x direction.

Similarly

→VBO=→ω×→rBO=5(−^k)×2(^i)=(−5×2)(^k×^i)

=10 m/s(−^j)=−10^j

→VCO=ω×→rCO=5(−^k)×2(^j)=−10(^k×^j)

=10(^i)(∵^k×^j=−^i)

→VDO=ω×→rDO=5(−^k)×2(−^i)=10×(^k×^i)

=10(^j)

[∵^k×^i=^j]

Hence, option D is incorrect.

As we know,

→VPQ=→ω×→rPQ

So here, →VAO=ω×→rAO

=5(−^k)×2(−^j)=−10^i

As we can see, the direction of velocity w.r.t O will be in negative x direction.

Similarly

→VBO=→ω×→rBO=5(−^k)×2(^i)=(−5×2)(^k×^i)

=10 m/s(−^j)=−10^j

→VCO=ω×→rCO=5(−^k)×2(^j)=−10(^k×^j)

=10(^i)(∵^k×^j=−^i)

→VDO=ω×→rDO=5(−^k)×2(−^i)=10×(^k×^i)

=10(^j)

[∵^k×^i=^j]

Hence, option D is incorrect.

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